The effect of structure on image classification using signatures

Abstract

Humans recognize transformed images from a very small number of samples. Inspired by this idea, we evaluate a classification method that requires only one sample per class, while providing invariance to image transformations generated by a compact group. This method is based on signatures computed for images. We test and illustrate this theory through simulations that highlight the role of image structure and sampling density, as well as how the signatures are constructed. We extend the existing theory to account for variations in recognition accuracy due to image structure.

Appendix A: Some concepts from group theory

A group is a set G together with an associative binary operation on G which has an identity and such that each element has an inverse with respect to the operation. That is,

• i. For all $a,b,c\in G$, it holds that $(a\cdot b)\cdot c=a\cdot (b\cdot c)$;

• ii. There exists an element e of G such that for all $a\in G$, $e\cdot a=a\cdot e=a$;

• iii. For every $a\in G$, there exists $b\in G$ such that $a\cdot b=b\cdot a=e$.

Example

The integers are a group with respect to the operation of addition. The identity element is 0, and the inverse of a is $-a$.

The action of a group G with identity e on a set X is a mapping from $G\times X$ to X,

$$\begin{aligned} G\times X\rightarrow X:\left( {g,x} \right) \rightarrow gx, \end{aligned}$$

such that

$$\begin{aligned} ex=x,\text { for all }x\in X, \end{aligned}$$

and

$$\begin{aligned} \left( {g_1 g_2 } \right) x=g_1 \left( {g_2 x} \right) \text { for all }g_1 ,g_2 \in G, x\in X. \end{aligned}$$

Example

Let X be a two-dimensional disk centered about the origin. Define the rotation $\phi$ as the operation which takes each point $\left( {r,\theta } \right)$ of the disk to the point $\left( {r,\theta +\phi } \right)$. The set of such rotations is a group which acts on X.

Suppose the group G acts on the set X. For each point $x\in X$, the orbit of x is the set $\left\{ {gx:g\in G} \right\}$.

Example

The orbit of the point $\left( {r,\theta } \right)$ in the disk example above is the circle of radius r about the origin.

A computation on the elements of a set X is said to be invariant under the action of the group G if the result of the computation on an element x is the same as the result of the element gx for all $g\in G$. In other words, if the results of the computation performed on any two elements of the orbit of x are indistinguishable.

Example

In the above disk example, consider the computation “distance from the origin.” The result of this computation is the same for all elements of any circle about the origin. That is, for all elements of the orbit of any point in the disk. Thus, this computation is invariant under the group action.

The simple examples above are given to illustrate the concepts. In the context of this paper, the set of interest will be a given set of images and the group action will be rotation of these images. The orbit of any particular image is in this case the set of all its rotations. A key idea is to find a computation (called a signature) which is invariant under this group action. This reduces the problem of distinguishing between different (potentially rotated) images to distinguishing between orbits of images, which is a much smaller set. This, in turn, drastically reduces the number of training samples required for certain image recognition tasks.

One final concept is needed. Given a group action G on a space X, the group G is said to be compact if its topology is compact. For example, if G is the set of two-dimensional rotations then G can be identified with the unit circle (if $\phi$ is an element of G, then $\phi$ can be identified with the element of the unit circle $e^{i\phi })$, and the unit circle is a compact subset of the plane.